Theorem ofcc 32074

Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcc.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcc.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ofcc.3	⊢ (𝜑 → 𝐶 ∈ 𝑋)

Assertion

Ref	Expression
ofcc	⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcc.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		fnconstg 6662	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4		ofcc.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		ofcc.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
6		fvconst2g 7077	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
7	1, 6	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
8	3, 4, 5, 7	ofcfval 32066	. 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
9		fconstmpt 5649	. 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))
10	8, 9	eqtr4di 2796	1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  {csn 4561   ↦ cmpt 5157   × cxp 5587   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   ∘_f/c cofc 32063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-ofc 32064

This theorem is referenced by: (None)